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We consider regular variation for marked point processes with independent heavy-tailed marks and prove a single large point heuristic: the limit measure is concentrated on the cone of point measures with one single point. We then investigate successive hidden regular variation removing the cone of point measures with at most k points, k ≥ 1, and prove a multiple large point phenomenon: the limit measure is concentrated on the cone of point measures with k + 1 points. We show how these results imply hidden regular variation in Skorokhod space of the associated risk process, in connection with the single/ multiple large point heuristic from (Ann. Probab. 47 (2019) 3551–3605). Finally, we provide an application to risk theory in a reinsurance model where the k largest claims are covered and we study the asymptotic behavior of the residual risk.

Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

Users of social networks display diversified behavior and online habits. For instance, a user’s tendency to reply to a post can depend on the user and the person posting. For convenience, we group users into aggregated behavioral patterns, focusing here on the tendency to reply to or reciprocate messages. The reciprocity feature in social networks reflects the information exchange among users. We study the properties of a preferential attachment model with heterogeneous reciprocity levels, give the growth rate of model edge counts, and prove the convergence of empirical degree frequencies to a limiting distribution. This limiting distribution is not only multivariate regularly varying, but also has the property of hidden regular variation.

In the context of multivariate regular variation, the authors establish the first-order asymptotics of the spectral risk measure of portfolio loss. Furthermore, by the notion of second-order regular variation, the second-order asymptotics of the spectral risk measure of portfolio loss is also presented. In order to illustrate the derived results, a numerical example with Monte Carlo simulation is carried out.

We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called ’normex’ approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles.

Stochastic volatility processes with heavy-tailed innovations are a well-known model for financial time series. In these models, the extremes of the log returns are mainly driven by the extremes of the i.i.d. innovation sequence which leads to a very strong form of asymptotic independence, that is, the coefficient of tail dependence is equal to 1/2 for all positive lags. We propose an alternative class of stochastic volatility models with heavy-tailed volatilities and examine their extreme value behavior. In particular, it is shown that, while lagged extreme observations are typically asymptotically independent, their coefficient of tail dependence can take on any value between 1/2 (corresponding to exact independence) and 1 (related to asymptotic dependence). Hence, this class allows for a much more flexible extremal dependence between consecutive observations than classical SV models and can thus describe the observed clustering of financial returns more realistically. The extremal dependence structure of lagged observations is analyzed in the framework of regular variation on the cone (0, ∞)d. As two auxiliary results which are of interest on their own we derive a new Breiman-type theorem about regular variation on (0, ∞)d for products of a random matrix and a regularly varying random vector and a statement about the joint extremal behavior of products of i.i.d. regularly varying random variables.

Data exhibiting heavy-tails in one or more dimensions is often studied using the framework of regular variation. In a multivariate setting this requires identifying specific forms of dependence in the data; this means identifying that the data tends to concentrate along particular directions and does not cover the full space. This is observed in various data sets from finance, insurance, network traffic, social networks, etc. In this paper we discuss the notions of full and strong asymptotic dependence for bivariate data along with the idea of hidden regular variation in these cases. In a risk analysis setting, this leads to improved risk estimation accuracy when regular methods provide a zero estimate of risk. Analyses of both real and simulated data sets illustrate concepts of generation and detection of such models.

Models based on assumptions of multivariate regular variation and hidden regular variation provide ways to describe a broad range of extremal dependence structures when marginal distributions are heavy tailed. Multivariate regular variation provides a rich description of extremal dependence in the case of asymptotic dependence, but fails to distinguish between exact independence and asymptotic independence. Hidden regular variation addresses this problem by requiring components of the random vector to be simultaneously large but on a smaller scale than the scale for the marginal distributions. In doing so, hidden regular variation typically restricts attention to that part of the probability space where all variables are simultaneously large. However, since under asymptotic independence the largest values do not occur in the same observation, the region where variables are simultaneously large may not be of primary interest. A different philosophy was offered in the paper of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497-546] which allows examination of distributional tails other than the joint tail. This approach used an asymptotic argument which conditions on one component of the random vector and finds the limiting conditional distribution of the remaining components as the conditioning variable becomes large. In this paper, we provide a thorough mathematical examination of the limiting arguments building on the orientation of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497-546]. We examine the conditions required for the assumptions made by the conditioning approach to hold, and highlight simililarities and differences between the new and established methods.

We study the behaviour of large values of extremal processes at small times, obtaining an analogue of the Fisher-Tippet-Gnedenko Theorem. Thus, necessary and sufficient conditions for local convergence of such maxima, linearly normalised, to the Fréchet or Gumbel distributions, are established. Weibull distributions are not possible limits in this situation. Moreover, assuming second order regular variation, we prove local asymptotic normality for intermediate order statistics, and derive explicit formulae for the normalising constants for tempered stable processes. We adapt Hill’s estimator of the tail index to the small time setting and establish its asymptotic normality under second order regular variation conditions, illustrating this with simulations. Applications to the fine structure of asset returns processes, possibly with infinite variation, are indicated.

Let X 1 , … , X n be independent nonnegative random variables with respective survival functions F ¯ 1 , … , F ¯ n , and let Θ 1 , … , Θ n be (not necessarily independent) nonnegative random variables, independent of X 1 , … , X n , satisfying certain moment conditions. This paper consists of two parts. In the first part, we investigate second-order expansions of 𝗣 ∑ i = 1 n X i > t as t → ∞ under the assumption that the F ¯ i are of second-order regular variation (2RV) with the same first-order index but with different second-order indexes. In the second part, under the assumption that the F ¯ 1 = ⋯ = F ¯ n have 2RV tails, second-order expansions of tail probabilities of the randomly weighted sum ∑ i = 1 n Θ i X i are studied. The closure property of 2RV under randomly weighted sum is also discussed. The main results in this paper generalize and strengthen several known results in the literature.